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Collinear Subtractions in Hadroproduction of Heavy Quarks

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B. A. Kniehl, G. Kramer, I. Schienbein, H. Spiesberger

High Energy Physics - Phenomenology · Submitted 21 Feb 2005 · 44 references

Pith's verdict: A careful, explicit worked reference that reproduces the authors' own earlier subtraction terms by an independent mass-factorization route; the one exception is disclosed but explained by attribution rather than calculation.

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arXiv IDhep-ph/0502194 (version 1)
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CitationB. A. Kniehl, G. Kramer, H. Spiesberger, I. Schienbein. "Collinear Subtractions in Hadroproduction of Heavy Quarks." arXiv:hep-ph/0502194 (2005). https://pith.science/paper/D4CXGB26
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The paper

The paper claims

The finite pieces that must be subtracted from a massive fixed-order calculation of heavy-quark hadroproduction are exactly the ones mass factorization predicts, generated by convoluting two-to-two partonic cross sections with order-$α_s$ パ

The reading

When a heavy quark is produced at large transverse momentum in a hadron collision, its mass $m$ stops being the only large scale and logarithms of $m^2/p_T^2$ have to be resummed. The standard route is to absorb those logarithms into parton distribution and fragmentation functions, leaving a hard cross section free of mass logarithms — a massive variable-flavour-number scheme. This paper asks whether the finite pieces that must be subtracted from the massive fixed-order calculation are precisely what mass factorization predicts, that is, whether they follow purely from convoluting ordinary two-to-two partonic cross sections with the order-$α_s$ perturbative PDFs and FFs. The authors perform that convolution explicitly for the $gg$, $q̄q$ and $gq$ channels and find term-by-term agreement with the subtraction terms they had previously extracted by taking the zero-mass limit of a fixed-order calculation and subtracting the genuine massless MS result, with a single exception in one coefficient of the gluon-fusion channel. If the picture holds, the matching terms are universal and the same bookkeeping can be reused for any other heavy-quark production process.

What carries the argument

The load-bearing object is the partonic factorization formula $dσ̃(a+b→Q+X)=f_{a→i}(x_1)⊗f_{b→j}(x_2)⊗dσ̂(i+j→k+X)⊗d_{k→Q}(z)$, expanded order by order in $α_s$. At NLO the hard cross section is the partonic one minus three convolutions, and those three convolutions are the subtraction terms. The work is in evaluating them: each convolution is reduced to a $dσ̂^{(0)}/d̂v$ times a Jacobian and a $δ$-function that fixes the integration variable, giving Eqs. (22), (29) and (36) for upper-line initial state, lower-line initial state and final state respectively. The universal ingredients are the order-$α_s$ MS splitting functions dressed with $łn(μ_F^2/m^2)$ and $łn(μ_F'^2/m^2)$ in Eqs. (11) and (12), and the final-state function $d^{(1)}_{Q→Q}(z̄)$ rewritten in Eq. (13) as a distribution in $v$ and $w$ with coefficients $A$, $B$, $C$ and $D$. That rewriting is what lets the convolution output be compared directly against the coefficient list $Δc_i$ of the earlier fixed-order subtraction.

Extended reading notes

Core claim

The central claim is that the collinear subtraction terms needed to build a massive variable-flavour-number scheme for one-particle inclusive heavy-quark hadroproduction are reproduced, coefficient by coefficient, by the mass-factorization convolutions $f^{(1)}_{a→i}⊗dσ̂^{(0)}$ and $dσ̂^{(0)}⊗d^{(1)}_{k→Q}$, where the partonic PDFs and FFs are the order-$α_s$ MS expressions with $m$ retained as a collinear regulator. Expanding the factorization formula to NLO yields three subtraction pieces — two initial-state and one final-state — and the authors evaluate them for $gg→Q̄Qg$, $q̄q→Q̄Qg$ and $gq→Q̄Qq$, keeping the heavy-quark mass terms inside the convoluted cross sections and then dropping them for the comparison. Written in the variables $v$ and $w$, each result decomposes into $δ(1-w)$, $1/(1-w)_+$, $(łn(1-w)/(1-w))_+$, $łn v$, $łn(1-w)$ and constant pieces with a definite colour structure split into Abelian and non-Abelian parts. Every one of these matches the corresponding term obtained earlier from $łim_{m→0}dσ̃(m)-dσ̂_{MS}$, except for an extra piece $Δc_1→Δc_1-C(s)C_A ǂrac{1}{9}v(1-v)$ in the $gg$ channel, which the authors attribute to heavy-quark loop contributions to the virtual corrections of the input massive calculation rather than to a missing term in the convolution. They also record, in the conclusions, that subtraction terms previously derived for the non-Abelian part of $γg→Q̄Qg$ and for $γq→Q̄Qq$ do not meet the mass-factorization expectation, and trace the latter to an error in the comparison calculation.

Load-bearing premise

The single mismatched coefficient in the gluon-fusion channel is assumed to be an artifact of heavy-quark loop contributions to the virtual corrections of the input massive calculation, an attribution the paper asserts without exhibiting the loop calculation that would demonstrate it.

Editorial extensions

If this is right

  • Any process whose subtraction terms are built from the same order-$α_s$ PDFs and FFs inherits the same finite matching pieces, so the scheme can be carried over to $γγ→Q̄Qg$, $γp→D^{⋆}+X$ and deep-inelastic heavy-quark production without recomputing them from scratch.
  • Because the subtraction terms carry the scale logarithms $łn(μ_F^2/m^2)$ and $łn(μ_F'^2/m^2)$ explicitly, the same expressions promote a fixed-scale result to an arbitrary fragmentation scale through the standard evolution equations.
  • Subtracting these terms from the massive fixed-order cross section leaves a hard part that reduces to the massless MS result as $m→0$ while retaining all $m^2/p_T^2$ terms, so the scheme interpolates continuously between the fixed-flavour and zero-mass descriptions.
  • The residual mismatch in the $gg$ channel, if confirmed to be an artifact of the input virtual corrections, would disappear from a recalculation, leaving the convolution picture exact at NLO.
  • The explicit colour decomposition into Abelian, $C_A^2$ and $C_A$-suppressed parts gives a template that an NNLO extension would need only supplement with the order-$α_s^2$ PDFs and FFs and the three-loop beta function.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The check is best read as a universality test rather than a derivation: it confirms that the same finite object appears along two very different calculational routes, which is what licenses using a single perturbative fragmentation function across processes.
  • The one unexplained coefficient sits exactly where heavy-quark loops enter the virtual amplitude; a targeted recalculation of that loop contribution would either remove the exception or show that mass factorization needs an extra, process-specific term at this order — a distinction that matters for any future extension beyond NLO.
  • The photon-initiated channels flagged in the conclusions are a natural stress test: if their subtraction terms genuinely fail to match the mass-factorization expectation, the likely cause is a scheme convention or an error in the comparison calculation rather than a breakdown of the convolution picture, and settling it would sharpen the definition of the massive scheme.
  • Because the procedure is mechanical — take the two-to-two cross sections, contract with the known splitting functions, and read off the coefficients — it suggests that matching terms for new processes could be generated largely automatically, lowering the cost of extending general-mass schemes to other reactions.
Original abstract

We present a detailed discussion of the collinear subtraction terms needed to establish a massive variable-flavour-number scheme for the one-particle inclusive production of heavy quarks in hadronic collisions. The subtraction terms are computed by convoluting appropriate partonic cross sections with perturbative parton distribution and fragmentation functions relying on the method of mass factorization. We find (with one minor exception) complete agreement with the subtraction terms obtained in a previous publication by comparing the zero-mass limit of a fixed-order calculation with the genuine massles results in the MSbar scheme. This presentation will be useful for extending the massive variable-flavour-number scheme to other processes.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,644 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.66 · human participants 0.01 · controlled comparison 0.14 · randomized assignment 0.01 · states quantitative result 0.11 · compares with prior results 0.97 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We present a detailed discussion of the collinear subtraction terms needed to establish a massive variable-flavour-number scheme for the one-particle inclusive production of heavy quarks in hadronic collisions. The subtraction terms are computed by convoluting appropriate partonic cross sections with perturbative parton distribution and fragmentation functions relying on the method of mass factorization. We find (with one minor exception) complete agreement with the subtraction terms obtained in a previous publication by comparing the zero-mass limit of a fixed-order calculation with the genuine massles results in the MSbar scheme. This presentation will be useful for extending the massive variable-flavour-number scheme to other processes.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We present a detailed discussion of the collinear subtraction terms needed to establish a massive variable-flavour-number scheme for the one-particle inclusive production of heavy quarks in hadronic collisions. contribution 1.00 0.91 0.14 asserted 0.99
s1 The subtraction terms are computed by convoluting appropriate partonic cross sections with perturbative parton distribution and fragmentation functions relying on the method of mass factorization. method 1.00 0.88 0.05 asserted 1.00
s2 We find (with one minor exception) complete agreement with the subtraction terms obtained in a previous publication by comparing the zero-mass limit of a fixed-order calculation with the genuine massles results in the MSbar scheme. result 1.00 0.92 0.51 asserted 1.00
s3 This presentation will be useful for extending the massive variable-flavour-number scheme to other processes. future work 0.60 0.13 0.13 hedged possibility 0.54

What kind of paper the classifier reports this is: theoretical result 0.95 · evidence from proof or derivation 0.91.

What was examined, and what was not

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. Feynman diagrams for the LO gluon-gluon fusion process
Figure 2. The LO quark-antiquark annihilation process
Figure 3. Sketch of kinematics of mass factorization for (a) upper
Figure 4. Feynman diagrams representing (a) f (1) g→g(x1) ⊗ dˆσ (0)(gg → QQ) and (b) f (1) g→g(x2)⊗dˆσ (0)(gg → QQ). The fermion loops on the external gluon lines are heavy-quark loops. The parts proportional to ln µ 2 F can be obtained by subtracting from the cross section in the gluon-gluon channel the term  f (1) g→g(x1) + f (1) g→g(x2)  ⊗ dˆσ (0)(g + g → Q + Q) (see
Figure 5. Feynman diagrams representing dˆσ (0)(gg → QQ) ⊗ d (1) Q→Q(z). to the function τ (v) = v 1 − v + 1 − v v + 4m2 sv(1 − v)  1 − m2 sv(1 − v)  , (45) which appears in the following expressions for the ∆ci . They are given by ∆c1 = 3 4 + ln v  ln µ ′ F 2 m2 + 1 − ln v − ln2 v  × 2C(s)CF τ (v)[CF − CAv(1 − v)] ,(46) ∆c2 =  ln µ ′ F 2 m2 − 1 − 2 ln v  × 2C(s)CF τ (v)[CF − CAv(1 − v)] , (47) ∆c3 = −2 × 2C(s)CF τ (v) [CF − CAv(1 − v)] , (48) ∆c5 = C(s)  C 2 F∆c qed 5 + 1 4 (C 2 A − 1)∆c oq 5  , (49) where ∆c qed 5 = 2v v1 −
Figure 6. Feynman diagrams representing of dˆσ (0)(gg → gg) ⊗ d (1) g→Q(z)
Figure 7. Feynman diagrams representing of f (1) g→Q(x1) ⊗ dˆσ (0)(Qg → Qg). The latter two relations can be derived from d (1) Q→Q(¯z) in Eq. (13) inspecting the expressions for B(v), C(v) and D(v, w). (Note that the parts with B(v) and C(v) contribute to ∆c11 and ∆c10, respectively, in cases where the 1/(1 − w)+ is canceled by factors (1 − w).) Now we turn to a comparison with the results in Ref. [12], which have been derived as described in Sec. 2. For µ ′ F = m (and neglecting terms of the order O(m2/s)), Eqs. (46)– (53) are in complete agreement with Eqs. (18) and (21)–(24) in Ref. [12].…
Figure 8. Feynman diagrams representing of f (1) g→Q(x2) ⊗ dˆσ (0)(gQ → Qg). ∆c qed 11 = 1 + v 2 vw (1 − 2w + 2w 2 ) ln µF 2 m2 , (55) ∆c oq 11 = 1 + v 2 v 2 1 2 w (1 − 2w + 2w 2 ) ln µF 2 m2 , (56) ∆c kq 11 = −∆c oq 11 . (57) f (1) g→Q(x2) ⊗ dˆσ (0)(gQ → Qg) Finally, we have a contribution from the u-channel cut in the initial state described by the diagram in
Figure 9. Feynman diagrams representing of dˆσ (0)(qq¯ → QQ) ⊗ d (1) Q→Q(z). dˆσ (0)(qq¯ → QQ) ⊗ d (1) Q→Q(z)
Figure 10. Feynman diagrams representing of dˆσ (0)(qq¯ → gg) ⊗ d (1) g→Q(z). dˆσ(qq¯ → gg) ⊗ d (1) g→Q(z), which will be given in the next subsection. dˆσ (0)(qq¯ → gg) ⊗ d (1) g→Q(z) The result for the cut diagram
Figure 11. Feynman diagrams representing of dˆσ (0)(gq → gq) ⊗ d (1) g→Q(z)
Figure 12. Feynman diagrams representing of f (1) g→Q(x1) ⊗ dˆσ (0)(Qq → Qq). ∆c cf 11 =  −2v 2 + 1 − 2v + 2v 2 2w + 2v 2w − v1v 2Y 2 + 3v − 2v 2 2Y  ln µ ′ F 2 m2 , (74) ∆c ca 11 =  −v 2 + v 2 (2 − 4v + 3v 2 )w v 2 1 + 2v 3 (1 − 2v)w 2 v 2 1 + 2v 4w 3 v 2 1 + v 2Y  ln µ ′ F 2 m2 . (75) Equations (74) and (75) are in agreement with Eqs. (69) and (70) in Ref. [12]. f (1) g→Q(x1) ⊗ dˆσ (0)(Qq → Qq) The contribution of the cut diagram
Figure 13. Feynman diagrams for the NLO gluon bremsstrahlung pro
Figure 14. Additional Feynman diagrams for the NLO gluon bremsstr
Figure 15. Feynman diagrams for the NLO gluon bremsstrahlung pro
Figure 16. Feynman diagrams for the NLO light-quark-initiated subp
Figure 17. Additional Feynman diagram for the NLO light-quark-initia

Figures reproduced from arXiv:hep-ph/0502194 by the authors.

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